The oscillation theorem for Tchebycheff spaces of bounded functions, and a converse

AbstractA (k + 1)-dimensional vector space U of real-valued functions defined on a subset of the real line is a Tchebycheff space (the linear space generated by a Tchebycheff system) if the number of zeros and the number of alternations in sign of each nonzero element of U is at most k. We show here that if U is a Tchebycheff space of bounded functions defined on a subset T of the real line, then for any pair of real-valued functions h0, h1 defined on T for which there is an element of U lying between h0 and h1 and bounded away from them, there exists an element of U that lies between h0 and h1 oscillates between them exactly k times. Additionally, a converse is given